Numerical Analysis of Nonlinear Shoaling and Its Impact on Suspended Sediment Dynamics across Surf and Swash Zones: A Navier–Stokes Approach Enhanced by Lagrangian Dynamic Smagorinsky Modeling with SPH

To develop a phase-resolving wave driver and establish a foundation for a comprehensive morphology model aimed at understanding the year-long circulation processes of sandy beaches and addressing beach erosion, the author introduced a wave driver based on the spatially averaged Navier–Stokes equations. Numerical investigations were conducted to evaluate the nonlinear shoaling characteristics of regular waves and their effects on suspended sediment dynamics across the surf and swash zones. To thoroughly validate the wave driver, bottom shear stress data from Sumer et al. (2013) were utilized, as bottom shear stress is a critical factor influencing the performance of morphology models. The author modeled the residual stress in the spatially averaged Navier–Stokes equations using the Lagrangian Dynamic Smagorinsky approach (Meneveau et al., 1996), which effectively resolves the turbulent flow of coherent structures—an essential feature in the surf and swash zones. Despite the importance of accurately capturing small-scale turbulent flows with coherent structures, many previous studies have relied on the standard Smagorinsky model, which is less effective in representing such turbulence. The author then numerically integrated the new wave driver using Smoothed Particle Hydrodynamics (SPH) with a Gaussian kernel function. The simulation successfully replicated complex wave dynamics, including severely deformed free water surfaces, free-falling water particles from wave crests, splashes upon particle impacts with the surface, and wave fingers formed by structured vortices on the up-wave side (Narayanaswamy and Dalrymple, 2002)—features that are notoriously challenging to replicate in computational fluid dynamics. Further analysis revealed that the widely used standard Smagorinsky model (\(\:{C}_{S}=0.12\)) excessively dampened the water surface profile due to overestimated energy dissipation from wave breaking. This led to the loss of critical flow features, such as reverse breaking, which are observed both in nature and in simulations using the Lagrangian Dynamic Smagorinsky model. Furthermore, instead of relying on the traditional quadratic friction law with a fixed friction coefficient, the author estimated bottom shear stress directly from the numerically simulated velocity profile and the dynamic Smagorinsky coefficient. These simulations showed that maximum bottom shear stress occurred when a broken wave, or bore, surged into the deep swash zone. The overall agreement with the measured data from Sumer et al. (2013) confirmed the accuracy and reliability of the new wave driver. The study demonstrated that the evolution of bottom shear stress within a wave period—particularly its asymmetric behavior in the surf zone, where most sediment movement occurs—can be accurately simulated using the new wave driver. These detailed characteristics of bottom shear stress are crucial for a morphology model that aims to capture the year-long circulation processes of sandy beaches and effectively address beach erosion. This is especially important because the seasonal migration of offshore bars is strongly influenced by asymmetrically accelerated flow and the resulting enhanced under-tow. The effectiveness of the newly proposed wave driver in capturing these key features, including boundary layer streaming, was further validated through numerical simulations, which demonstrated the evolution of suspended sediment across the surf and swash zones, with a sediment cloud gradually drifting offshore due to the intensified under-tow.